The oscilloscope will show the voltage difference between the tip of the probe and the ground of the probe. Most scopes have 1 MΩ resistance between tip and ground, so the voltage you are measuring is affected the same as putting a 1 MΩ resistor between the two points you are connecting tip and ground to.
For many circuits, 1 MΩ is high enough to not alter the voltage much. For example, CMOS digital logic signals have impedances of 10s to 100s of Ohms typically. 1 MΩ on even a 1 kΩ source is only going to cause a 0.1% voltage change.
For cases where 1 MΩ is too low, you can switch the probe to "10x" mode. This makes its impedance 10x higher, so 10 MΩ. That's good enough for most circuits. That's also the typical resistance of a voltmeter.
There is one extra wrinkle for line powered (as apposed to battery powered) scopes. The scope chassis is usually connected to the line ground, which should be connected to earth ground in the building you are in. That means you can't just put a scope probe anywhere. You have to consider that the probe ground clip is connected to earth ground.
This generally doesn't matter for circuits on your desk running from isolated power supplies. It's a real issue when trying to measure anything line-connected. This is one reason isolation transformers are often used when working on line-connected equipment in a lab.
Your diagram actually shows this. Note how one side of the AC line and the scope chassis are connected to ground. It also shows a 1:1 isolation transformer. The stuff connected to the right side of the isolation transformer can be at arbitrary DC potential relative to the left side. Put another way, the stuff on the right side is floating. You can connect any one point you choose to earth ground. This is what is being done by connecting the scope probe ground to the junction between Z1 and Z2. That point is now at earth ground.
When floating, any one connection between some point on the circuit and ground won't cause current to flow. Once you make that one connection though, the circuit is no longer floating, and all other voltage are now relative to ground. That means you can get hurt by touching some other point and ground at the same time. You have to be aware that connecting the scope ground clip uses up the one free connection to the rest of the world, and that circuit now has dangerous ground-referenced voltages. If you touch some other part of the circuit and a metal radiator, the chassis of your computer, the scope, or anything else grounded, you can get zapped.